Energy level

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https://en.wikipedia.org/wiki/Energy_level 

 

Energy levels for an electron in an atom: ground state and excited states. After absorbing energy, an electron may "jump" from the ground state to a higher energy excited state.

A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The term is commonly used for the energy levels of the electrons in atoms, ions, or molecules, which are bound by the electric field of the nucleus, but can also refer to energy levels of nuclei or vibrational or rotational energy levels in molecules. The energy spectrum of a system with such discrete energy levels is said to be quantized.

In chemistry and atomic physics, an electron shell, or principal energy level, may be thought of as the orbit of one or more electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" (also called "K shell"), followed by the "2 shell" (or "L shell"), then the "3 shell" (or "M shell"), and so on farther and farther from the nucleus. The shells correspond with the principal quantum numbers (n = 1, 2, 3, 4 ...) or are labeled alphabetically with letters used in the X-ray notation (K, L, M, N...).

Each shell can contain only a fixed number of electrons: The first shell can hold up to two electrons, the second shell can hold up to eight (2 + 6) electrons, the third shell can hold up to 18 (2 + 6 + 10) and so on. The general formula is that the nth shell can in principle hold up to 2n² electrons. Since electrons are electrically attracted to the nucleus, an atom's electrons will generally occupy outer shells only if the more inner shells have already been completely filled by other electrons. However, this is not a strict requirement: atoms may have two or even three incomplete outer shells. (See Madelung rule for more details.) For an explanation of why electrons exist in these shells see electron configuration.

If the potential energy is set to zero at infinite distance from the atomic nucleus or molecule, the usual convention, then bound electron states have negative potential energy.

If an atom, ion, or molecule is at the lowest possible energy level, it and its electrons are said to be in the ground state. If it is at a higher energy level, it is said to be excited, or any electrons that have higher energy than the ground state are excited. An energy level is regarded as degenerate if there is more than one measurable quantum mechanical state associated with it.

Explanation

Wavefunctions of a hydrogen atom, showing the probability of finding the electron in the space around the nucleus. Each stationary state defines a specific energy level of the atom.

Quantized energy levels result from the wave behavior of particles, which gives a relationship between a particle's energy and its wavelength. For a confined particle such as an electron in an atom, the wave functions that have well defined energies have the form of a standing wave. States having well-defined energies are called stationary states because they are the states that do not change in time. Informally, these states correspond to a whole number of wavelengths of the wavefunction along a closed path (a path that ends where it started), such as a circular orbit around an atom, where the number of wavelengths gives the type of atomic orbital (0 for s-orbitals, 1 for p-orbitals and so on). Elementary examples that show mathematically how energy levels come about are the particle in a box and the quantum harmonic oscillator.

Any superposition (linear combination) of energy states is also a quantum state, but such states change with time and do not have well-defined energies. A measurement of the energy results in the collapse of the wavefunction, which results in a new state that consists of just a single energy state. Measurement of the possible energy levels of an object is called spectroscopy.

History

The first evidence of quantization in atoms was the observation of spectral lines in light from the sun in the early 1800s by Joseph von Fraunhofer and William Hyde Wollaston. The notion of energy levels was proposed in 1913 by Danish physicist Niels Bohr in the Bohr theory of the atom. The modern quantum mechanical theory giving an explanation of these energy levels in terms of the Schrödinger equation was advanced by Erwin Schrödinger and Werner Heisenberg in 1926.

Atoms

Intrinsic energy levels

In the formulas for energy of electrons at various levels given below in an atom, the zero point for energy is set when the electron in question has completely left the atom, i.e. when the electron's principal quantum number n = ∞. When the electron is bound to the atom in any closer value of n, the electron's energy is lower and is considered negative.

Orbital state energy level: atom/ion with nucleus + one electron

Assume there is one electron in a given atomic orbital in a hydrogen-like atom (ion). The energy of its state is mainly determined by the electrostatic interaction of the (negative) electron with the (positive) nucleus. The energy levels of an electron around a nucleus are given by :

${\displaystyle E_n=-hc{R}_{\mathrm{\infty }}\frac{Z^2}{n^2}}$

(typically between 1 eV and 10³ eV), where R_∞ is the Rydberg constant, Z is the atomic number, n is the principal quantum number, h is Planck's constant, and c is the speed of light. For hydrogen-like atoms (ions) only, the Rydberg levels depend only on the principal quantum number n.

This equation is obtained from combining the Rydberg formula for any hydrogen-like element (shown below) with E = h ν = h c / λ assuming that the principal quantum number n above = n₁ in the Rydberg formula and n₂ = ∞ (principal quantum number of the energy level the electron descends from, when emitting a photon). The Rydberg formula was derived from empirical spectroscopic emission data.

${\displaystyle \frac{1}{\lambda }=R{Z}^2\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)}$

An equivalent formula can be derived quantum mechanically from the time-independent Schrödinger equation with a kinetic energy Hamiltonian operator using a wave function as an eigenfunction to obtain the energy levels as eigenvalues, but the Rydberg constant would be replaced by other fundamental physics constants.

Electron-electron interactions in atoms

If there is more than one electron around the atom, electron-electron-interactions raise the energy level. These interactions are often neglected if the spatial overlap of the electron wavefunctions is low.

For multi-electron atoms, interactions between electrons cause the preceding equation to be no longer accurate as stated simply with Z as the atomic number. A simple (though not complete) way to understand this is as a shielding effect, where the outer electrons see an effective nucleus of reduced charge, since the inner electrons are bound tightly to the nucleus and partially cancel its charge. This leads to an approximate correction where Z is substituted with an effective nuclear charge symbolized as Z_eff that depends strongly on the principal quantum number.

$${\displaystyle E_{n,\ell }=-hc{R}_{\mathrm{\infty }}\frac{{Z_{\mathrm{e}\mathrm{f}\mathrm{f}}}^2}{n^2}}$$

In such cases, the orbital types (determined by the azimuthal quantum number ℓ) as well as their levels within the molecule affect Z_eff and therefore also affect the various atomic electron energy levels. The Aufbau principle of filling an atom with electrons for an electron configuration takes these differing energy levels into account. For filling an atom with electrons in the ground state, the lowest energy levels are filled first and consistent with the Pauli exclusion principle, the Aufbau principle, and Hund's rule.

Fine structure splitting

Fine structure arises from relativistic kinetic energy corrections, spin–orbit coupling (an electrodynamic interaction between the electron's spin and motion and the nucleus's electric field) and the Darwin term (contact term interaction of s shell electrons inside the nucleus). These affect the levels by a typical order of magnitude of 10⁻³ eV.

Hyperfine structure

Main article: Hyperfine structure

This even finer structure is due to electron–nucleus spin–spin interaction, resulting in a typical change in the energy levels by a typical order of magnitude of 10⁻⁴ eV.

Energy levels due to external fields

Zeeman effect

Main article: Zeeman effect

There is an interaction energy associated with the magnetic dipole moment, μ_L, arising from the electronic orbital angular momentum, L, given by

${\displaystyle U=-{\mathit{\mu }}_L\cdot \mathbf{B}}$

with

${\displaystyle -{\mathit{\mu }}_L={\displaystyle \frac{e\hslash }{2m}}\mathbf{L}={\mu }_B\mathbf{L}}$.

Additionally taking into account the magnetic momentum arising from the electron spin.

Due to relativistic effects (Dirac equation), there is a magnetic momentum, μ_S, arising from the electron spin

${\displaystyle -{\mathit{\mu }}_S=-{\mu }_B{g}_S\mathbf{S}}$,

with g_S the electron-spin g-factor (about 2), resulting in a total magnetic moment, μ,

${\displaystyle \mathit{\mu }={\mathit{\mu }}_L+{\mathit{\mu }}_S}$.

The interaction energy therefore becomes

${\displaystyle U_B=-\mathit{\mu }\cdot \mathbf{B}={\mu }_{B}B(M_L+g_S{M}_S)}$.

Stark effect

Main article: Stark effect

Molecules

Chemical bonds between atoms in a molecule form because they make the situation more stable for the involved atoms, which generally means the sum energy level for the involved atoms in the molecule is lower than if the atoms were not so bonded. As separate atoms approach each other to covalently bond, their orbitals affect each other's energy levels to form bonding and antibonding molecular orbitals. The energy level of the bonding orbitals is lower, and the energy level of the antibonding orbitals is higher. For the bond in the molecule to be stable, the covalent bonding electrons occupy the lower energy bonding orbital, which may be signified by such symbols as σ or π depending on the situation. Corresponding anti-bonding orbitals can be signified by adding an asterisk to get σ* or π* orbitals. A non-bonding orbital in a molecule is an orbital with electrons in outer shells which do not participate in bonding and its energy level is the same as that of the constituent atom. Such orbitals can be designated as n orbitals. The electrons in an n orbital are typically lone pairs.  In polyatomic molecules, different vibrational and rotational energy levels are also involved.

Roughly speaking, a molecular energy state, i.e. an eigenstate of the molecular Hamiltonian, is the sum of the electronic, vibrational, rotational, nuclear, and translational components, such that:

$${\displaystyle E=E_{\text{electronic}}+E_{\text{vibrational}}+E_{\text{rotational}}+E_{\text{nuclear}}+E_{\text{translational}}}$$

where E_electronic is an eigenvalue of the electronic molecular Hamiltonian (the value of the potential energy surface) at the equilibrium geometry of the molecule.

The molecular energy levels are labelled by the molecular term symbols. The specific energies of these components vary with the specific energy state and the substance.

Energy level diagrams

There are various types of energy level diagrams for bonds between atoms in a molecule.

Examples

Molecular orbital diagrams, Jablonski diagrams, and Franck–Condon diagrams.

Energy level transitions

Further information: atomic electron transition and molecular electron transition

 

An increase in energy level from E₁ to E₂ resulting from absorption of a photon represented by the red squiggly arrow, and whose energy is h ν

 

A decrease in energy level from E₂ to E₁ resulting in emission of a photon represented by the red squiggly arrow, and whose energy is h ν

Electrons in atoms and molecules can change (make transitions in) energy levels by emitting or absorbing a photon (of electromagnetic radiation), whose energy must be exactly equal to the energy difference between the two levels. Electrons can also be completely removed from a chemical species such as an atom, molecule, or ion. Complete removal of an electron from an atom can be a form of ionization, which is effectively moving the electron out to an orbital with an infinite principal quantum number, in effect so far away so as to have practically no more effect on the remaining atom (ion). For various types of atoms, there are 1st, 2nd, 3rd, etc. ionization energies for removing the 1st, then the 2nd, then the 3rd, etc. of the highest energy electrons, respectively, from the atom originally in the ground state. Energy in corresponding opposite quantities can also be released, sometimes in the form of photon energy, when electrons are added to positively charged ions or sometimes atoms. Molecules can also undergo transitions in their vibrational or rotational energy levels. Energy level transitions can also be nonradiative, meaning emission or absorption of a photon is not involved.

If an atom, ion, or molecule is at the lowest possible energy level, it and its electrons are said to be in the ground state. If it is at a higher energy level, it is said to be excited, or any electrons that have higher energy than the ground state are excited. Such a species can be excited to a higher energy level by absorbing a photon whose energy is equal to the energy difference between the levels. Conversely, an excited species can go to a lower energy level by spontaneously emitting a photon equal to the energy difference. A photon's energy is equal to Planck's constant (h) times its frequency (f) and thus is proportional to its frequency, or inversely to its wavelength (λ).

ΔE = h f = h c / λ,

since c, the speed of light, equals to f λ

Correspondingly, many kinds of spectroscopy are based on detecting the frequency or wavelength of the emitted or absorbed photons to provide information on the material analyzed, including information on the energy levels and electronic structure of materials obtained by analyzing the spectrum.

An asterisk is commonly used to designate an excited state. An electron transition in a molecule's bond from a ground state to an excited state may have a designation such as σ → σ*, π → π*, or n → π* meaning excitation of an electron from a σ bonding to a σ antibonding orbital, from a π bonding to a π antibonding orbital, or from an n non-bonding to a π antibonding orbital. Reverse electron transitions for all these types of excited molecules are also possible to return to their ground states, which can be designated as σ* → σ, π* → π, or π* → n.

A transition in an energy level of an electron in a molecule may be combined with a vibrational transition and called a vibronic transition. A vibrational and rotational transition may be combined by rovibrational coupling. In rovibronic coupling, electron transitions are simultaneously combined with both vibrational and rotational transitions. Photons involved in transitions may have energy of various ranges in the electromagnetic spectrum, such as X-ray, ultraviolet, visible light, infrared, or microwave radiation, depending on the type of transition. In a very general way, energy level differences between electronic states are larger, differences between vibrational levels are intermediate, and differences between rotational levels are smaller, although there can be overlap. Translational energy levels are practically continuous and can be calculated as kinetic energy using classical mechanics.

Higher temperature causes fluid atoms and molecules to move faster increasing their translational energy, and thermally excites molecules to higher average amplitudes of vibrational and rotational modes (excites the molecules to higher internal energy levels). This means that as temperature rises, translational, vibrational, and rotational contributions to molecular heat capacity let molecules absorb heat and hold more internal energy. Conduction of heat typically occurs as molecules or atoms collide transferring the heat between each other. At even higher temperatures, electrons can be thermally excited to higher energy orbitals in atoms or molecules. A subsequent drop of an electron to a lower energy level can release a photon, causing a possibly colored glow.

An electron farther from the nucleus has higher potential energy than an electron closer to the nucleus, thus it becomes less bound to the nucleus, since its potential energy is negative and inversely dependent on its distance from the nucleus.

Crystalline materials

Crystalline solids are found to have energy bands, instead of or in addition to energy levels. Electrons can take on any energy within an unfilled band. At first this appears to be an exception to the requirement for energy levels. However, as shown in band theory, energy bands are actually made up of many discrete energy levels which are too close together to resolve. Within a band the number of levels is of the order of the number of atoms in the crystal, so although electrons are actually restricted to these energies, they appear to be able to take on a continuum of values. The important energy levels in a crystal are the top of the valence band, the bottom of the conduction band, the Fermi level, the vacuum level, and the energy levels of any defect states in the crystal.

Hydraulic analogy

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Hydraulic_analogy 

 

Analogy between a hydraulic circuit (left) and an electronic circuit (right).

The electronic–hydraulic analogy (derisively referred to as the drain-pipe theory by Oliver Lodge) is the most widely used analogy for "electron fluid" in a metal conductor. Since electric current is invisible and the processes in play in electronics are often difficult to demonstrate, the various electronic components are represented by hydraulic equivalents. Electricity (as well as heat) was originally understood to be a kind of fluid, and the names of certain electric quantities (such as current) are derived from hydraulic equivalents. As with all analogies, it demands an intuitive and competent understanding of the baseline paradigms (electronics and hydraulics).

Paradigms

There is no unique paradigm for establishing this analogy. Two paradigms can be used to introduce the concept to students using pressure induced by gravity or by pumps.

In the version with pressure induced by gravity, large tanks of water are held up high, or are filled to differing water levels, and the potential energy of the water head is the pressure source. This is reminiscent of electrical diagrams with an up arrow pointing to +V, grounded pins that otherwise are not shown connecting to anything, and so on. This has the advantage of associating electric potential with gravitational potential.

A second paradigm is a completely enclosed version with pumps providing pressure only and no gravity. This is reminiscent of a circuit diagram with a voltage source shown and the wires actually completing a circuit. This paradigm is further discussed below.

Other paradigms highlight the similarities between equations governing the flow of fluid and the flow of charge. Flow and pressure variables can be calculated in both steady and transient fluid flow situations with the use of the hydraulic ohm analogy. Hydraulic ohms are the units of hydraulic impedance, which is defined as the ratio of pressure to volume flow rate. The pressure and volume flow variables are treated as phasors in this definition, so possess a phase as well as magnitude.

A slightly different paradigm is used in acoustics, where acoustic impedance is defined as a relationship between acoustic pressure and acoustic particle velocity. In this paradigm, a large cavity with a hole is analogous to a capacitor that stores compressional energy when the time-dependent pressure deviates from atmospheric pressure. A hole (or long tube) is analogous to an inductor that stores kinetic energy associated with the flow of air.

Hydraulic analogy with horizontal water flow

Voltage, current, and charge

In general, electric potential is equivalent to hydraulic head. This model assumes that the water is flowing horizontally, so that the force of gravity can be ignored. In this case, electric potential is equivalent to pressure. The voltage (or voltage drop or potential difference) is a difference in pressure between two points. Electric potential and voltage are usually measured in volts.

Electric current is equivalent to a hydraulic volume flow rate; that is, the volumetric quantity of flowing water over time. Usually measured in amperes.

Electric charge is equivalent to a quantity of water.

Basic circuit elements

• 

  Conducting wire:  a simple hose.

• 

  Resistor: a constricted pipe.

• 

  Node in Kirchhoff's junction rule: A pipe tee filled with flowing water.

A relatively wide hose completely filled with water is equivalent to a conducting wire. A rigidly mounted pipe is equivalent to a trace on a circuit board. When comparing to a trace or wire, the hose or pipe should be thought of as having semi-permanent caps on the ends. Connecting one end of a wire to a circuit is equivalent to un-capping one end of the hose and attaching it to another. With few exceptions (such as a high-voltage power source), a wire with only one end attached to a circuit will do nothing; the hose remains capped on the free end, and thus adds nothing to the circuit.

A resistor is equivalent to a constriction in the bore of a pipe which requires more pressure to pass the same amount of water. All pipes have some resistance to flow, just as all wires and traces have some resistance to current.

A node (or junction) in Kirchhoff's junction rule is equivalent to a pipe tee. The net flow of water into a piping tee (filled with water) must equal the net flow out.

• 

  Capacitor:  a flexible diaphragm sealed inside a pipe.

• 

  Inductor:  a heavy paddle wheel or turbine placed in the current.

• 

  Voltage or current source:  A dynamic pump with feedback control.

A capacitor is equivalent to a tank with one connection at each end and a rubber sheet dividing the tank in two lengthwise (a hydraulic accumulator). When water is forced into one pipe, equal water is simultaneously forced out of the other pipe, yet no water can penetrate the rubber diaphragm. Energy is stored by the stretching of the rubber. As more current flows "through" the capacitor, the back-pressure (voltage) becomes greater, thus current "leads" voltage in a capacitor. As the back-pressure from the stretched rubber approaches the applied pressure, the current becomes less and less. Thus capacitors "filter out" constant pressure differences and slowly varying, low-frequency pressure differences, while allowing rapid changes in pressure to pass through.

An inductor is equivalent to a heavy paddle wheel placed in the current. The mass of the wheel and the size of the blades restrict the water's ability to rapidly change its rate of flow (current) through the wheel due to the effects of inertia, but, given time, a constant flowing stream will pass mostly unimpeded through the wheel, as it turns at the same speed as the water flow. The mass and surface area of the wheel and its blades are analogous to inductance, and friction between its axle and the axle bearings corresponds to the resistance that accompanies any non-superconducting inductor.

An alternative inductor model is simply a long pipe, perhaps coiled into a spiral for convenience. This fluid-inertia device is used in real life as an essential component of a hydraulic ram. The inertia of the water flowing through the pipe produces the inductance effect; inductors "filter out" rapid changes in flow, while allowing slow variations in current to be passed through. The drag imposed by the walls of the pipe is somewhat analogous to parasitic resistance. In either model, the pressure difference (voltage) across the device must be present before the current will start moving, thus in inductors, voltage "leads" current. As the current increases, approaching the limits imposed by its own internal friction and of the current that the rest of the circuit can provide, the pressure drop across the device becomes lower and lower.

An ideal voltage source (ideal battery) or ideal current source is a dynamic pump with feedback control. A pressure meter on both sides shows that regardless of the current being produced, this kind of pump produces constant pressure difference. If one terminal is kept fixed at ground, another analogy is a large body of water at a high elevation, sufficiently large that the drawn water does not affect the water level. To create the analog of an ideal current source, use a positive displacement pump: A current meter (little paddle wheel) shows that when this kind of pump is driven at a constant speed, it maintains a constant speed of the little paddle wheel.

Other circuit elements

• 

  A simple one-way ball-type check valve, in its "open" state acts as a diode in its conducting state.

• 

  A pressure-actuated valve combined with a one-way check valve acts as a (field-effect) transistor.

• 

  Like a one-way check valve, a diode blocks current that flows the wrong way. Current that flows the right way goes through almost unchanged.

• 

  A simple A/C circuit consisting of an oscillating pump, a "diode" valve, and a "capacitor" tank. Any kind of motor could be used here to drive the pump, as long as it oscillates.

A diode is equivalent to a one-way check valve with a slightly leaky valve seat. As with a diode, a small pressure difference is needed before the valve opens. And like a diode, too much reverse bias can damage or destroy the valve assembly.

A transistor is a valve in which a diaphragm, controlled by a low-current signal (either constant current for a BJT or constant pressure for a FET), moves a plunger which affects the current through another section of pipe.

CMOS is a combination of two MOSFET transistors. As the input pressure changes, the pistons allow the output to connect to either zero or positive pressure.

A memristor is a needle valve operated by a flow meter. As water flows through in the forward direction, the needle valve restricts flow more; as water flows the other direction, the needle valve opens further, providing less resistance.

Practical application

On the basis of this analogy Johan van Veen developed around 1937 a method to compute tidal currents with an electric analogue. After the North Sea flood of 1953 in The Netherlands he elaborated this idea, which eventually lead to the analog computer ‘’Deltar’’, which was used to make the hydraulic computations for the closures in the framework of the Delta Works.

Principal equivalents

EM wave speed (velocity of propagation) is equivalent to the speed of sound in water. When a light switch is flipped, the electric wave travels very quickly through the wires.

Charge flow speed (drift velocity) is equivalent to the particle speed of water. The moving charges themselves move rather slowly.

DC is equivalent to the a constant flow of water in a circuit of pipes.

Low frequency AC is equivalent to water oscillating back and forth in a pipe

Higher-frequency AC and transmission lines is somewhat equivalent to sound being transmitted through the water pipes, though this does not properly mirror the cyclical reversal of alternating electric current. As described, the fluid flow conveys pressure fluctuations, but fluids do not reverse at high rates in hydraulic systems, which the above "low frequency" entry does accurately describe. A better concept (if sound waves are to be the phenomenon) is that of direct current with high-frequency "ripple" superimposed.

Inductive spark used in induction coils is similar to water hammer, caused by the inertia of water

Equation examples

Some examples of analogous electrical and hydraulic equations:

type	hydraulic	electric	thermal	mechanical
quantity	volume ${\displaystyle V}$ [m3]	charge ${\displaystyle q}$ [C]	heat ${\displaystyle Q}$ [J]	momentum ${\displaystyle P}$ [Ns]
quantity flux	Volumetric flow rate ${\displaystyle {\mathrm{\Phi }}_V}$ [m3/s]	current ${\displaystyle I}$ [A=C/s]	heat transfer rate ${\displaystyle \dot{Q}}$ [J/s]	velocity ${\displaystyle v}$ [m/s=J/Ns]
flux density	velocity ${\displaystyle v}$ [m/s]	current density ${\displaystyle j}$ [C/(m2·s) = A/m²]	heat flux ${\displaystyle {\dot{Q}}^{″}}$ [W/m2]	stress ${\displaystyle \sigma }$ [N/m2 = Pa]
potential	pressure ${\displaystyle p}$ [Pa=J/m3=N/m2]	potential ${\displaystyle \varphi }$ [V=J/C=W/A]	temperature ${\displaystyle T}$ [K]	force ${\displaystyle F}$ [N]
linear model	Poiseuille's law ${\displaystyle {\mathrm{\Phi }}_V=\frac{\pi r^4}{8\eta }\frac{\mathrm{\Delta }{p}^{\star }}{\ell }}$	Ohm's law ${\displaystyle j=-\sigma \mathrm{\nabla }\varphi }$	Fourier's law ${\displaystyle {\dot{Q}}^{″}=\kappa \mathrm{\nabla }T}$	dashpot ${\displaystyle \sigma =c\mathrm{\Delta }v}$

If the differential equations have the same form, the response will be similar.

Limits to the analogy

If taken too far, the water analogy can create misconceptions. For it to be useful, one must remain aware of the regions where electricity and water behave very differently.

Fields (Maxwell equations, inductance): Electrons can push or pull other distant electrons via their fields, while water molecules experience forces only from direct contact with other molecules. For this reason, waves in water travel at the speed of sound, but waves in a sea of charge will travel much faster as the forces from one electron are applied to many distant electrons and not to only the neighbors in direct contact. In a hydraulic transmission line, the energy flows as mechanical waves through the water, but in an electric transmission line the energy flows as fields in the space surrounding the wires, and does not flow inside the metal. Also, an accelerating electron will drag its neighbors along while attracting them, both because of magnetic forces.

Charge: Unlike water, movable charge carriers can be positive or negative, and conductors can exhibit an overall positive or negative net charge. The mobile carriers in electric currents are usually electrons, but sometimes they are charged positively, such as the positive ions in an electrolyte, the H⁺ ions in proton conductors or holes in p-type semiconductors and some (very rare) conductors.

Leaking pipes: The electric charge of an electrical circuit and its elements is usually almost equal to zero, hence it is (almost) constant. This is formalized in Kirchhoff's current law, which does not have an analogy to hydraulic systems, where the amount of the liquid is not usually constant. Even with incompressible liquid the system may contain such elements as pistons and open pools, so the volume of liquid contained in a part of the system can change. For this reason, continuing electric currents require closed loops rather than hydraulics' open source/sink resembling spigots and buckets.

Fluid velocity and resistance of metals: As with water hoses, the carrier drift velocity in conductors is directly proportional to current. However, water only experiences drag via the pipes' inner surface, while charges are slowed at all points within a metal, as with water forced through a filter. Also, typical velocity of charge carriers within a conductor is less than centimeters per minute, and the "electrical friction" is extremely high. If charges ever flowed as fast as water can flow in pipes, the electric current would be immense, and the conductors would become incandescently hot and perhaps vaporize. To model the resistance and the charge-velocity of metals, perhaps a pipe packed with sponge, or a narrow straw filled with syrup, would be a better analogy than a large-diameter water pipe.

Quantum mechanics: Solid conductors and insulators contain charges at more than one discrete level of atomic orbit energy, while the water in one region of a pipe can only have a single value of pressure. For this reason there is no hydraulic explanation for such things as a battery's charge pumping ability, a diode's depletion layer and voltage drop, solar cell functions, Peltier effect, etc., however equivalent devices can be designed which exhibit similar responses, although some of the mechanisms would only serve to regulate the flow curves rather than to contribute to the component's primary function.

In order for the model to be useful, the reader or student must have a substantial understanding of the model (hydraulic) system's principles. It also requires that the principles can be transferred to the target (electrical) system. Hydraulic systems are deceptively simple: the phenomenon of pump cavitation is a known, complex problem that few people outside of the fluid power or irrigation industries would understand. For those who do, the hydraulic analogy is amusing, as no "cavitation" equivalent exists in electrical engineering. The hydraulic analogy can give a mistaken sense of understanding that will be exposed once a detailed description of electrical circuit theory is required.

One must also consider the difficulties in trying to make an analogy match reality completely. The above "electrical friction" example, where the hydraulic analog is a pipe filled with sponge material, illustrates the problem: the model must be increased in complexity beyond any realistic scenario.